Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones

TL;DR

This paper presents explicit polynomial-sized semidefinite representations for hyperbolicity cones related to elementary symmetric polynomials and derivatives of spectrahedral cones, enabling efficient optimization over these convex cones.

Contribution

It introduces new polynomial-sized semidefinite representations for hyperbolicity cones of elementary symmetric polynomials and their derivatives, extending to general spectrahedral cones.

Findings

Explicit polynomial-sized semidefinite representations are constructed.

These representations enable solving linear programs over the cones.

The approach applies to dual cones and facilitates semidefinite programming.

Abstract

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in $k,m$, and $n$) of the hyperbolicity cones associated with $k$th directional derivatives of polynomials of the form $p(x) = \det(\sum_{i=1}^{n}A_i x_i)$ where the $A_i$ are $m\times m$ symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.